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Matter supermultiplet coupled to supergravity in the second order canonical formalism
ANNALS
OF PHYSICS
203, 207-228 (1990)
Matter Supermultiplet Coupled to Supergravity in the Second Order Canonical Formalism A. FOUSSATS Facultad U.N.R.
AND
0. ZANDRON*
de Ciencias Exactas Ingenieria y Agrimensura, Av., Pellegrini 250, 2000 Rosario, Argentina
Received December 27. 1989
Starting from the first order canonical covariant formalism (CCF) for the Wesss Zumino @ SUGRA coupled system, we find the second order canonical vierbein formalism (CVF). The primary constraints obtained in the CCF, are analyzed in the framework of the second order CVF. From the total Hamiltonian density obtained in the CCF and by making the space-time decomposition, the Hamiltonian in the CVF is found. This is the correct Hamiltonian which generates the time evolution of generic functional of fields and momenta. It remains defined by means of the set of the first class constraints of the system. This set closes the well-known constraint algebra. #n 1990 Academic Press, Inc.
1.
INTRODUCTION
In a previous paper [l], the canonical covariant formalism (CCF) on group manifold for the supersymmetric Wess-Zumino matter multiplet coupled to supergravity (W-Z) 0 SUGRA was constructed. As it was already shown in several works [2-81, the first order CCF is particularly useful for treating systems which are gauge invariant under some group of transformations. When the CCF is applied to such systems the general result obtained is that all the constraints are primary (there are no secondary constraints in the formalism) and none of them is first class. This relevant characteristic together with the covariance of the formalism in all their steps, allows us to obtain the constraints, the total Hamiltonian, and the consistency condition for the constraints in a very simple way. Thus the equations of motion and the rheonomic conditions are obtained by means of straightforward algebraic manipulations. In particular, the correct Hamiltonian as a first class dynamical quantity remains given unambiguously once the primary constraints are computed from the Lagrangian density. The total Hamiltonian thus defined is strongly conserved, i.e., dH=O.
On the other hand, it is well known that the Hamiltonian description of the gravitational field by means of vierbein and spin connection [g-14] is more * Fellows of the Consejo National de Investigaciones Cientilicas y Tecnicas, Argentina.
207 0003-4916190 $7.50 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved
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adequate than that given through the metric tensor and the metric affine connection [ 15, 161. In the framework of the canonical vie&in formulation (CVF) for gravity and supergravity, both the first [ 141 and the second [lo] order formalisms must also be distinguished. In the second order formalism for pure supergravity, the spin connection is a dependent functional of the vierbein and gravitino fields. The construction of the second order formalism starting directly from the first order CVF involves very heavy algebraic manipulations [13, 141. These difficulties are even enough, when a supersymmetric system (i.e., Yang-Mills or Wess-Zumino supermultiplets) coupled to supergravity is considered. Moreover, it is well known that the transition from the first order formalism to the second order one, allows us to separate the true dynamical degrees of freedom from those of gauge degrees of freedom. Thus the importance of treating these constrained systems in the second order Hamiltonian formalism, grows up when the quantization of such systems is considered. Also this kind of formulation should be useful for implementing new ideas about the quantization of the gravitational field. At this stage the CCF plays an important role, because of its more simple and compact structure. In particular the CCF can be used as an interesting formal mechanism to derive and analyze the set of constraints in CVF. Really, the CCF does not define a standard mechanical system in the sense that it is not a proper Hamiltonian theory as it is in the case of the CVF which propagates data defined on an initial hypersurface C [9]. Consequently, the first order CCF and the CVF must be related, and it is possible to show that this relation is non-trivial [4, 71. The first question is that in the construction of the canonical covariant formalism, the form-brackets are introduced and they must be related to the usual Poisson brackets defined in the CVF. Besides, it must be pointed out that the CCF takes the exterior derivative d as a form-observable and it has no direct analogue in the CVF. Thus, in the CCF the first class dynamical quantity defined as the Hamiltonian density, is not the Hamiltonian which generates the time evolution of generic functionals of fields and momenta. Therefore, starting from the bosonic 4-form HT defined in [ 11, we must construct the generator of the time evolution. This must be done by finding the set of first class constraints which are the generators of the constraint algebra [lo] and the Hamiltonian remains given as a functional of these generators. As we will see, in practice the CVF can be recovered from the CCF by taking into this last formalism some field equations of motion as constraints. For simple supergravity, the first order CVF was directly constructed [13, 14) and it was also found starting from the CCF [4]. Both results were confronted and the agreement and differences were pointed out. Recently [21], starting from the CCF we have constructed the second order CVF for the super Yang-Mills theory coupled to supergravity in dimension D = 4. In this paper we have treated the W-Z@ SUGRA coupled system, because it constitutes an interesting example for the application of the CCF on group manifold. In this system the first order formalism, requires the introduction of two kind of fields. The geometrical l-form gauge fields to describe the supergravity
209
SECONDORDER CANONICALFORMALISM
sector and a set of non-geometrical O-form fields for the description of the Wess-Zumino matter supermultiplet. Moreover, this coupled system allows us to introduce in the CCF the definition of supercurrent of the W-Z matter multiplet which acts as a source for the supergravitational field. It also allows us to obtain information about the set of supergravity auxiliary fields. As in this system the torsion equation is different from zero, this equaton gives rise to a modified expression for the Lorentz spin connection. As we will see later on, the spin connection obtained from the torsion equation is a functional of the vierbein, the gravitino, and the Majorana spinor O-form field 2, partner of the boson fields A, B which are necessary to describe the W-Z supermultiplet. Finally, we comment that the purpose of this kind of treatment is to give a second order component formalism for non-linear supergravities in the Riemann curvature, starting from the CCF on group manifold previously developed by us (see Refs. [17-191). At least for the linear case it should be important to confront the results obtained from the CCF with those given in Ref. [20], for the interesting SYM @ SUGRA N= 1 D = 10 coupled system [ 173. The paper is organized as follows: In Section 2, we briefly summarize the definitions used and the results obtained in the construction of the CCF for the W-Z @ SUGRA coupled system. In Section 3, we make the space-time decomposition in M4 = G/H and we give the relation between the form-brackets and the Poisson brackets. In Section 4, starting from the CCF we construct the second order CVF by writing in components all the quantities which in the CCF are written in form language. The general expressions of the formalism restricted to the three-dimensional surface ,E are found, and the primary constraints obtained in the CCF are analyzed in the framework of the second order CVF. In Section 5, we explicitly compute the set of first class constraints which verify the constraint algebra. Using these constraints, the Hamiltonian which generates the time evolution of a generic functional of the fields and momenta remains defined. Finally in Section 6, we analyze how the Dirac brackets for the system under consideration can be obtained.
2. THE CCF FOR W-Z@SUGRA
SYSTEM: DEFINITIONS
AND RESULTS
We summarize the relevant results obained in Ref. [l] in the construction of the lirst order CCF on a group manifold G, for W-Z@ SUGRA constrained coupled system. The dynamics is described by introducing the set of pseudo-connection l-forms pA = (c@, V”, 5”) and a set of three O-forms. The set of l-forms are the gauge SUGRA fields and the three O-form fields are necessary to describe the supersymmetric W-Z multiplet (0, 0, 4). These O-forms dynamical fields are A, B (bosonic objects), and A (anticommuting object), transforming respectively as a scalar, a pseudo-scalar, and a Majorana spinor. The invariance under global supersymmetric transformation laws and the rheonomic symmetry conditions in a first order formalism require the introduction of two O-form independent dynamical fields U”
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and W“ transforming respectively as a vector and axial vector under the Lorentz group. Therefore, the first order CCF for this coupled system is constructed by starting from the set of eight dynamical fields namely: cPb, Vu, <“, A, B, A, U”, W“ and we will denote their corresponding canonical conjugate momenta by ‘~t,~, x,, 71x0 7q‘4), QY), $3 z,, Ea. The Lagrangian density for the system allows us to obtain the eight primary constraints given by @ab=71,bM0
(2.la)
@a =lt n -22WbCA Vd&abed -0 -
(2.lb)
@” = 710L - 4(y,y,5)”
(2.lc)
@(A)=
I’+,+
uavb
+$iB([r, @(B,=~C(B,+
A V” z50 h
~“5) wavb
A
-@([Ay”<)
v’ A
V,zO
v’
A
A
y=$+iyUnVb
vd&,bcd-6&‘~~ob~
A
A V’A
A
vu
A
vb
(2.ld)
I/dE,bcd-
6i/iz,bc
A T/”
A
vb
(2.le)
v,E:o
Vd&abrdzO
(2.lf)
(Pa=ToXO
Wg) (2.lh)
q, = E, Fz0.
From the canonical Hamiltonian given in Ref. [ 1, Eq. (3.4)] and the constraints (2.1), we can define the total massive Hamiltonian as ff,(WZ) = H,,,(m) +z’
+ /tub A @jab+ A0 A y’,+c”
A Cpa+a”
A 0,
+/I”
A @a + ~4~~)
A @CA1
+ /itB)
A Q(B)
A va,
(2.2)
where H,,,(m)
= H;,U,GR* + H~,Z(O) + H,WZ.
(2.3)
The Hamiltonian pieces @zGRA, Hz:(O), and HF were explicitly written in Eqs. (3.5), (3.6), and (3.10) of Ref. [l], respectively. We have shown, that the total Hamiltonian (2.2) is the correct first class dynamical quantity which describes the dynamics of the constrained coupled system under consideration. Moreover, we remember that in the CCF all the constraints are primary (there are no secondary constraints in the formalism) and none of them is first class (see Ref. [l, Eq. (4.1)]. When the fundamental equation &I = (A, HT) + aA, involving the form-brackets for a generic form A is used for the constraints, obtain the following results:
(2.4) we
SECOND
(i)
For the compound
ORDER
CANONICAL
FORMALISM
211
index A = (ab, a, c(), the equations, d@,=(@,,
H,)zO.
(2.5)
These are the equation of motion for the supergravity sector (see Ref. [ 1, Eqs. (4.6)]. In the inhomogeneous supergravitational field equations (4.6) of Ref. [l], the quantity J= (Jab, J,, J,) 3-form multiplet appears which is the Osp(4/1) supercurrent of the W-Z matter multiplet acting as source for the supergravitational field. In the framework of the CCF they remain defined by
Jab= (@uh,H,W=)
(2.6a)
J, = (@,, H,W=)
(2.6b)
25= (cs, fly=)
(2.6~)
and they can be written as a function of the set of non-minimal auxiliary fields for supergravity (A,, Ai, Y, 9, x) and the spinor-vector j, as we have shown in [l]. In particular, the axial vector A, is defined by A, = 32y5 y,ll
and therefore the component
(2.7)
Jab of the supercurrent can be written as Jab=;A,V”
A V, A V,,.
(2.8 1
From the torsion equation d@ab = (@ab, HT) ~00,
(2.9)
we obtain R”=
-LPbcdAbV~ 8
A V,.
(2.10)
(ii) For the compound index M which takes into account the five O-forms dynamical fields, the equations d@,=(@,,H,)=O,
(2.11)
give rise to the five Eqs. (4.3) of Ref. [l] for the W-Z sector in the massive case. Those equations determine the dynamics of the W-Z system and from them the following rheonomic conditions for the fields A, B, and A.can be obtained: dA=
U,V”-;5~
dB= W,V”-i;iy,<
Gal= x, vu - i( iy, U” + ysya wa) 5.
(2.12a) (2.12b) (2.12c)
Equations (2.12) show the rheonomic symmetry of the total Hamiltonian (2.2) and they also make it possible to turn the first order Hamiltonian formalism into
212
FOUSSATS
AND
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the second order one. Moreover, from Eqs. (2.12) we can see that the O-forms bosonic fields U” and W” introduced above and the Majorana spinor-vector field X, represent the spatial derivative of the W-Z fields A, B, and I, respectively. All other results about CCF that we wll use in what follows are given in Ref. [ 11. 3. THE SPACE-TIME
DECOMPOSITION AND THE RELATION BETWEEN POISSON AND FORM BRACKETS
With the purpose of constructing the second order Hamiltonian formalism for the system under consideration, we will follow a similar procedure to that given in [3,4] for gravity and pure supergravity. We use also our previous behaviours, notations, and conventions given in the paper [21]. The first step is to carry out the space-time decomposition. Subsequently, all the equations and quantities which we have been writing in form language using exterior calculus on differential forms, must be written in components. We will use for the Greek and Latin indices the same convention we have used in [21, Section 2.23. All the dynamical fields must be considered only as reduced forms, that is, they are forms defined on the coset manifold M4 = G/H (physical superspace), where H is a subgroup of the Lie supergroup G and it is the exact gauge symmetry group. Afterward, the extended forms can be obtained naturally for the whole supergroup G, through H gauge transformations [22]. Moreover, we consider that the reduced forms defined on M4 are written in the holonomic basis dxP. Finally, equations, fields and forms must be projected on a space-like x0 = t = to hypersurface 2 of three dimensions. This requires the introduction of the injection map x: C -+ M4, in such a way that the associated pull-back x* acts on any generic forms by setting t = to and dt = 0. For the space-time decomposition in M4 we follow Ref. [15]. The vierbein V” = Lf dx” is split according to the expression (2.11) of Ref. [21]. Besides, we use, the notation given in Eqs. (2.12t(2.15) and the properties given in Appendix C of Ref. [4] for the alternating tensor sUbrd in tangent space and the alternating tensor siik on ,X. We use the above mentioned expressions of Refs. [21,4] whenever they are necessary. We call p;(x) = [ --~II(~‘;~ (x), L:(x),
- “*Tc’,(x)E~~ dx’ A dxk = g- %;(x)&,
where the 2-form zi is defined on the hypersurface C.
(3.1)
SECOND
ORDER
CANONICAL
213
FORMALISM
Analogously, for the other O-form dynamical fields CL””= (A, B, 1, U”, IV’) the corresponding canonical conjugate momenta rr,,,,= (rccA,,Q), 8, r,, E,) are 3-forms, therefore we have the definitions, 71M=sg
1
-“‘I~,(x)E~~
dxi A dx’ A dxk = ZZ,(x)Q,
= n,,,,(x) d3x, (3.2)
g-“2
where the volume 3-form Q, is defined on Z and n,(x) = (n,,,(x), Z~,,,(X), e(x), K(X)? &b)). For the constraints (2.1), similar expressions as those given in (3.1) and (3.2) for the momenta hold. Then, the usual Poisson brackets between the components of the fields and canonical conjugate momenta are given by [L”;(x),
74(y)]
= - [Q’(y),
L”;(x)]
= 6”&J3(x,
y)
Cmiab(x),~jd(Y)I = - C~(‘,db), oi”“(x)l = Sab[cd,cw3(X,y) Cti(X)9
n/?j(Y)l
=
Cna’(Y),
gZi(x)]
=6”/j6/63(X,
y)
(3.3a) (3.3b) (3.3c)
cm)? ~,(.Y)l = C&(Y), ax)1 = QB3(x, y)
(3.3d)
[uu(x), 6,(y)] = - [%(v), vu(x)] = b”b~3(-%y) CA(x), n,,,(Y)1 = - L-n,,,(Y)? A(x)1 =63(x, Y)
(3.3e)
[ w”(xh
Eb(Y)l
=
-
[&b(Y),
w”(x)l
=
&?j3(x,
ev)
[B(x), n,,,(y)1 = - [n,,,(y), B(x)1 = J3(x, Y).
(3.3f) Wg) (3.3h)
The Dirac delta function appearing in the above equations is defined by
b3(x, ev)f(x) =fb), sd3x
(3.4)
without appeal to the metric. As was briefly commented in the introduction there is another question to take into account to relate both formalisms the CCF and the CVF. In the CCF the form-brackets are defined and they play an important role in the construction of the Hamiltonian equations of motion and in the determination of the properties of the constraints. Such form-brackets must be related to the Poisson brackets defined in CVF. As was shown in [4] the Poisson brackets yields more information than the form brackets. Moreover, a particular subset of Poisson brackets can be related to the form-brackets by means of an integral relationship. Taking into account the properties for the form-brackets given in Eqs. (2.3) and (2.4) of Ref. [l], we are going to write for supergravity an analogue integral relationship to that given in [4] for gravity. We assume that the generic forms A(x) and B(y) at the points x and y, respec-
214
FOUSSATS
AND
ZANDRON
tively, on the hypersurface C, are polynomials in the fields and momenta variables only. The forms A(x) and B(y) have a and b degrees and Fermi grading IA 1 and IBl, respectively. The Poisson brackets between forms [A(x), B(y)] define a form of degree a+ b on Cx Z. Because of the way they were defined, the Poisson brackets between forms do not contain derivatives of the Dirac distribution and they can be evaluated by expanding the forms A(x) and B(y) in the holonomic bases dx’, dy’ and then using the Poisson brackets (3.3) between fields and momenta components. Thus we can write the following Poisson brackets between dynamical canonical conjugate (forms) variables:
CP”(X),~Ay)l=;
hA, g-1’2(~)~&) hi * &’ * dykd3(x, Y)
=P,g-l’*(y)dxj
A C,(Y)~~(X,
y)
(3Sa)
and b”(x),
n,(Y)1
1 =- 3! d”Ng - “*( Y)E~~( y) dy’ A dy’ A dyk d3(x, y) = P,
g-1’*(y)i2,63(x,
(3.5b)
y).
The Poisson brackets between generic forms A, B, C of degrees a, b, c and Fermi grading IA 1, 1B(, and ICl, respectively, remain defined by means of the definition (3.5) and they verify the following properties: [A(x),
B(y)]
= (- l)“b+ 1+‘A”E’ [B(y),
A(x)]
(3.6a)
CA(x), B(Y) * C(Y)] = [4x), B(Y)] * C(Y) +(-l)“b+‘A”B’B(y) A [A(x), C(y)]
(3.6b)
CA(x) * B(x), C(y)1 = A(x) * L-W), C(Y)] + (- l)b’+‘B”C’ [A(x), C(y)] A B(x)
(3.6~)
(- l)“‘+‘“““[A(x), +(-I)
[B(y),
C(z)]]
cb+‘C”e’ CC(z), [A(x), B(y)11
(- l)b”+ ‘We’ [B(y),
[C(z),
A(x)]]
= 0.
(3.6d)
Now, the integral relationship which allows us to relate the form-backets subset of Poisson brackets between forms is given by
with a
A CA(x), B(Y)] A KY),
(3.7)
t-11 a+l+‘A”B’jzor(x)
A (A, B) A /3(x)=/jzXZa(x)
SECOND
ORDER
CANONICAL
215
FORMALISM
where CI and/I are two test forms of degrees 3 --a and 3 - 6, respectively, and arbitrary Fermi grading. We can check (3.7) for pairs of canonical variables. If we consider that the form-brackets for pairs of canonical variables is [ 1 ] (/iA, 7c8)= (-l)o+‘+‘A’8AB,
(3.8)
the left hand side of (3.7) is equal to LiA,J, LX(X) A b(x). Taking (3.5) for the Poisson brackets, the right hand side of (3.7) is
the expressions
4x) A CPA(X)3 7-c&)1 A P(Y) 1.j zxz a(x) A dx’ A dy’ A dyk A b,(y)
=
hA,
jz
a(x)
A
dy'g-1'2(y)&,~(~)~3(X,y)
dx’ J: sz,6!B,(Y)g-“2(Y)~3(x,
=dA, s,dx)Adx’s, Ph4S3(x, Y) d3y
= dA,
Y)
j
4x1
z
A P(x).
(3.9)
This last result and the property (3.6a) of commutation for the Poisson brackets, shows that the form-bracket definition and the definition of the Poisson brackets for the components of the canonical conjugate variables are consistent.
4. THE SECOND ORDER CVF FROM THE CCF
To find explicitly the second order canonical component considering the following expression for the torsion R”=dV“-wab
A
i’-,-$[A
formalism,
we begin by (4.1)
~“5,
associated to the graded Poincare algebra and the solution (2.10) for the torsion provided by the torsion field equation (2.9). From these equations it is possible to obtain the solution for the l-form spin connection gab of the W-Z@ SUGRA system, whose components in the holonomic basis are generally given by G/y
v, 5, A) = zY/b( V) - qb(&
A).
(4.2)
(4)
Here we write hrab( V) t o indicate that such a functional is the same functional of the vierbein as in pure gravity. The second piece of the right hand side of (4.2) is the component of the contorsion tensor C, which is a functional of the gravitino field 5 and the axial vector A, defined in (2.7). To find the explicit expression for the contorsion tensor we proceed as follows. 595/203/l-15
216
FOUSSATS
AND
ZANDRON
As it is well known the full covariant derivative including world (metric) connection satisfies the metricity condition
both the spin and the
i?,L", + (FipL,, -v-plrvLap = 0,
(4.3)
where 4P’ TVis the affme connection of the four-dimensional manifold M4 and is no longer symmetric in its lower indices because of the torsion generated by the spinor fields 5 and 1. On the other hand, by considering the spin connection (:)Ob and the affine connection 3r:k (no longer symmetric in his lower indices) on the three surface L’, it can also be written as
akLq + 'A>L,-
3rkj~ai = 0,
(4.4)
by the vierbein postulate on the three-dimensional surface Z. Moreover, by multiplying Eq. (4.4) by the normal n, to the surface C we obtain the equation
d,n"+'i>n,=O.
(4.5)
Now, we consider the restriction to the three-dimensional surface Z of the torsion expression (4.1) rewritten in components and the explicit expression (2.10) for the torsion R". Thus we obtain
a,,Lq, + !2$ L,, - t[ckyyjl Taking
+ ~~~~~~~~~~~~~ = 0.
(4.6)
into account Eq. (4.3) and Eq.(4.6) we can find
sp, E g4rpkj- “rpjk) = f[,,yp
(4.7)
We note that in the second orderC4FVF EqQ4.3) and (4.4) allow us to determine completely both spin connections mpab and mkab. After some algebraic manipulations, the following relationship between the components of both connections can be found: (ziab = (:)pb + (nbLaj - naLbj)Kii. In (4.8) K, is the extrinsic curvature of the three-dimensional manifold M4 and it is defined by
Ku=
~(-g,,+N,,,j+N,,I,)-cji,.
(4.8) surface C in the
(4.9)
The double stroke 11in (4.9) denotes the covariant derivative on the three-surface C, only including the aftine connection.
SECOND
The contorsion
ORDER
CANONICAL
217
FORMALISM
tensor C is defined by (4.10)
CPjl” = - W,,” + svpp + S,,“) and the component
C,, is given by Co, = -f(giYj5,-riY15j+5jYi51)-~&iikAk
(4.11a)
Ciik=-~(giYj5k-giYk4j+~jYiSk)+$EijkAi.
(4.1 lb)
From Eq. (4.8) we can see that the information provided by Eqs. (4.3) and (4.4) are quite different. As the affrne connection 3ri(jkJ in (4.4) is a functional of the metric ‘gY on C and therefore of the Lai only, then (%)kabresults as a functional of the Loi and its spatial derivative only. While, from Eq. (4.3) the restriction to the three-dimensional surface z of the affrne connection 4Ti(jk) and the spin connection (4)
&kub on M4 are functionals of both the components Lai of the vierbein and the corresponding canonical conjugate momenta through the extrinsic curvature K,,, as we will see later on. Of course, the complete expression for the two spin connections involved in (4.8) also differ in their respective parts containing the contorsion. We note that Eqs. (4.8) and (4.9) are formally identical to the corresponding equations given in Ref. [3] for the pure simple supergravity. As we can see from (4.11) in the W-Z @ SUGRA coupled system, the difference is contained in the components C,, and C, of the contorsion tensor C. Such components are not only bilinear fuctionals of the gravitino field 5 as in pure supergravity, but they also contain the spinor field A of the W-Z matter supermultiplet. That is to say, in the system under consideration the contorsion C as well as the torsion S are generated by the two spinor fields which are present in the theory. Now we are going to analyze the primary constraints (2.1) obtained in CCF. We begin by making a brief commentary about the constraints. The possibility of using different Lagrangian densities means that there is not a unique set of canonical conjugate momenta and consequently there is not a unique set of primary constraints in the CCF. For example, if we start from the following Lagrangian density for supergravity dP=Rab
A V’ A Vdc abrd
+ 4p
A Y5Yatf
A
va/“,
(4.12)
which differs from a total exterior derivative of our starting Lagrangian density, we cannot obtain the primary constraints (2.la) and (2.lb) but rather we obtain sob
=
n,b
-
@,=71,x0.
v’
A
Vd&,brd
% 0
(4.13a) (4.13b)
Moreover, by looking at the second order formalism [9, lo] we see that the momentum components dab are strongly equal to zero. Therefore, when both formalisms are related and we go from the CCF to the second order CVF, it is
218
FOUSSATS
AND
ZANDRON
made evident that it was a good choice to start from the Lagrangian density which provides Q, z 0. Another question to take into account is that the CCF plays, with respect to the first order CVF, an analogous role to that played by the CVF of first order with respect to the second order CVF. Thus we will consider that all the primary constraints in the CCF remain at least weakly zero in the CVF. Therefore, in our case we are going to consider that the constraints (2.la), (2.lg), and (2.lh) are strongly equal to zero: Qab = n,b = 0,
(Pn=Ta=O 7
q,=&,=o.
(4.14)
On the other hand, we also assume that the restrictions to C of the constraints (2.lc), (2.ld), (2.le), and (2.lf) are strongly equal to zero, so that the following equations hold: X*w = 0,
x*q,,
= 0,
x*@(B) = 0,
p!P=o.
(4.15)
For the remaining constraint Q,, which is a functional of the spin connection we consider the restriction to Z and take it as a weakly zero quantity, that is, x*cDa E *o x 0.
(4.16)
In the second order CVF, (4.15) correspond to second class constraints, which can be eliminated from the theory by defining the modified Dirac brackets. As we will see below these new Dirac brackets can be evaluated by starting from the second class constraints. Here Eqs. (4.15) allow us to solve for the corresponding momenta K,, 7ccaj, rccBj, and 9,. Equations (4.14~(4.16) among others, show that all weakly zero quantities in the CCF remain at least weakly zero in the CVF as it was commented above. Now, we must compute explicitly $,. Using for the 2-form $. an analogous expression to that given in (3.1), after some algebra we arrive at *a=g-‘/**&=
17,+4(L,kg”-L,‘g’k)Kkj~i+4n,CiikgikCi,
where the explicit expression for the component t/$; = rr,i + 4g’12(Lakgij - Loigjk)Kkj According to the available momentum 2-form 17,:
literature
(4.17)
$=’ is + 4g”*n,C;,
gj”.
(4.18)
[3, 133, we have defined the following
(3)
n, = n, + 2 dfd h vb&,bcd = n, + 2(:)cdA Vb&&cd + 4n, c:k gjkci.
(4.19)
SECOND
ORDER
CANONICAL
219
FORMALISM
(3) We note that the functional which was denoted by bed in (4.19) is the same functional of the vierbein T/” like that of pure gravity without spinor terms. Thus, the Z7, definition is formally identical to that given in Ref. [3]. Again, from (4.19) we can see that the information about t&e Wess-Zumino mater supermultiplet is contained in both, the spin connection gcd on ,X and the components C, of the contorsion tensor C. The importance of Eq. (4.19) is clearly shown because it relates the new momentum 2-form Ii’, with the momentum n, defined in the CCF. From Eq. (4.19) an expression relating the components 17,’ and .i can be easily written. In what follows we will use the momentum Z7, to find the remaining quantities in the second order CVF. Now, it is possible to define the following antisymmetric weakly vanishing quantities Mab : Mobd3x=$,
A V,-$,
A V,zO.
(4.20)
Starting from the CCF, these quantities appear directly as primary constraints because they are functionals of the constraints tin which are the restriction to z of the primary constaints Qa. These six constraints Mnb are the generators of the local Lorentz group for all the fields of the theory and in the CVF they are first class constraints [lo]. Moreover, as we can see from (4.20) they must be considered as densities. Using Eq. (4.18) and performing some algebraic manipulations, the generators Mab can be written as M,, = +(IZ,‘Lbi - IliLa;)
+ 2(n, Lbi - n,Lai) Cik gjkgl/*
-I- 2(L,‘Lbk - L,‘Lak)Cjk,
(4.21)
g’?
Now, we must define other quantities which will be useful later on. If we call # the symmetric tensor density canonical conjugate momentum of the three metric g,, it can be written as .u
E ;(n,iLaj
+ n,jLai)
=
-g1/2(~(f)
_ gtjK),
(4.22)
where ZP’ is the symmetric part of the extrinsic curvature given in (4.9) and K is the trace. These relations involving K” and the momentum rc’i hold, Kc”‘=
-g-1/2(n”-
igFlt)
(4.23a)
2
K= $‘3 ~~K..-K2=g-1(n”x..J’
where Kcyl is the antisymmetric
(4.23b) 0
part of Ku.
ln2)+KCGl~ 2
,_
[Jrlr
(4.23~)
220
FOUSSATS
Before concluding
AND
ZANDRON
this section we give the set of useful relations, M’k
=
LaiLbkM
(4.24a)
nb
Mil
= -L”$,~M,~
(4.24b)
nbnbi = 2~‘~ + 4p’c,‘,
gjk
(4.24~)
Lbinbj = 4,.&i _ MJ _ 4gWcUill which we will use in the explicit computation
5. THE FIRST CLASS CONSTRAINTS
(4.24d)
9
of the first class constraints.
AND THE HAMILTONIAN
Now, we must find the Hamiltonian 2 for the W-ZOSUGRA system, as generator of the time evolution for generic functional of field and momentum variables. The rate of change in time of any functional F of the canonical variables is given by the Poisson bracket of F with this total Hamiltonian, i’= [F, $1.
(5.1)
From the bosonic 4-form (2.2) provided by the CCF, we first make the space-time decomposition of M 4. Next, by choosing the time variable so that the l-form dx” can be detached, we have
[H,(m)=J’dx”A 2 and we consider the remaining bosonic 3-form dimensional surface C. The Hamiltonian 2 turns out to be of the form,
(5.2)
2
integrated
in the three-
x&(X) +L”&%(x) +2(x)&l] d3X, where & are the time components of the l-form field #. To compute explicitly &(x) we consider the expression (2.2) and (2.3). explicit expression for H,,,(m) was given in Ref. [l, Eqs. (3.5), (3.6), (3.10)]. write the piece of HT containing the constraints we use Eqs.(4.14) and (4.15) we also take into account Eqs. (3.8) of Ref. [l] for the Lagrange multipliers. Subsequently, to construct the expression (5.3) we write explicitly the component of all the forms present in HT. After some algebraic manipulations
(5.3)
The To and time and
SECOND
ORDER
CANONICAL
221
FORMALISM
by neglecting total divergences, it is possible to arrive at the following expressions for the 3-forms J&(X) d3x: A?a,d3x=2{(-2RC~
V’E,~,~+J~~)+$[@~A
Sad3x={(-2Rb”/t
Vde,,,.,+4~~/\y,y,~+J,) AVd+ob,]
- hkd %‘d3x=
{(-8LG&y, - ih”
A
cDu>
(54a)
V,]}lzzO
(54b)
A @h}lrzo
V”+4[y5yu
A
T/,-c@~A
A
R”+25) (5.4c)
(,zO.
We note that the expressions enclosed in the parentheses in (5.4a, b, c) are the inhomogeneous equations for the supergravitational field with source (see Ref. [ 11). Thus, Eqs. (5.4) clearly show that .&,, &, and s%?are all weakly zero quantities and they are the first class constraints of the coupled system under consideration. The field equation of motion and the rheonomic equations in the CCF are obtained by imposing the conditions (2.5) and (2.11) on the constraints. These equations are the covariant extension of the preservation of the constraints in time demanded in the CVF. Therefore, starting from the CCF it is possible to see how certain constraints are given directly by the field equations of motion, as was already commented on in the introduction. In the CVF analogous constraints appear when the preservation of the constraints in time is demanded [13]. The results obtained in (5.4) must be expected and for the pure supergravity sector we can compare our results with those previously obtained in Refs [3,4] and we can check the consistency of both results. Now, we must write in components the expressions (5.4) for ZA(x) restricted to the three-dimensional surface C. We begin by computing the 3-form J& d3x. The second order formalism implicates the elimination of the spin connection as an independent variable. That is to say, we must use the explicit expression for the torsion RaIz and solve it for the spin connection. This is equivalent to taking the equation 2R”
A
as strongly equal to zero. Taking
VdEabcd
-
Juh = 0,
into account the definition
&,d3x=1C/,
A
V,-t+bb
A
(5.5)
(4.20) we find
V,s14,,d3xz0.
To compute *U d3x, the initial step is the use of the Gauss-Codacci [23]. Thus the first piece of ZU d3x can be written as 2Rb”
A
VdE,bcd
=
(5.6)
equations
[4n,( ‘R + K2 - K”K,,) + 8Lak( gvKjkili - K,,/,)] g”2 d3X.
(5.7)
222
FOUSSATS
AND
To evaluate the second piece containing
ZANDRON
9[ we use
where
gi[j=aifj-
$$;b[jr*b.
(5.9)
The covariant derivative on the three-surface Z is given by (5.10) therefore the following relation holds: gi
$(nbLak - n”Lbk)Kiktjrab.
(5.11)
To write in components the 3-form J,, we consider for it the explicit expression given in Eq. (5.4b) of Ref. [ 11. That expression contains &I, dB, and 9A. In the second order formalism we must replace these quantities by their rheonomic solutions (2.12). Finally, we must consider the expression (4.18) for the constraints $0’ and by using Eqs. (4.21)-(4.24) after a long and rather heavy algebra it is possible to write Xa in components. Thus, by considering the decomposition LaoXa= N,c%?’ +Nkxk
(5.12a)
L; J, = NI J’ + NkJk
(5.12b)
we arrive at ‘/23R+,g-9,‘+2
P=4(-g -1
128 g
1’2AiAi-
‘712)+~g1’2giy15’5iy15j
&g1/2([jy1(i)AkEijk
+ ;g1’2Di[jysy’~kc’ik} +J’+ajMi’+MU(-~giyItli+~Ak&,jk) zk = gni kllr.+ gg’/zg kJ.(-i[+ 2 ei+ leiilA ) I 8 1 II - 2g”2Dj[iysy,~k& -
2i~kYjri~nii+g1’2(riyj5k)(riYIrj)
+
$e’Y’~k&,,k+
+ +ajMjk+
(5.13)
$(Digk - D,&)x’
Jk +M”($&y,<,+
$iIEuk)
+MiI[g-1’2(nik--gikn)+f~k-,y.5i-~A’&ki,],
(5.14)
SECOND
ORDER
CANONICAL
223
FORMALISM
where J’ =g”*{
-6(U,
U’ + W, W’) + $A’[iyk
+ 6iDjxyjl-
$AkAk
-i(lJi-~~i)[6&,r’k~j-~B[jyl~k]~vk + ( Wi-isXy,<,)[6rITikcj+
iiA[jy,ck]Evk
+ 3m(mA2 +mB* + 212) - 6im&A + iysB)y’<, + 3imti(A Jk=g1’*{6;i(U’
+ iy,B)y,f’k~jEYk} +iys W’)r,+
(5.15) fAk[iy,
$A,[iyk[j~Q’
-6iD,xy.il-6(U,-/&)U,-6(W,-i;zy,t,)W, -i(Ui-@i)[6~y,rk,~j+$B[jyk<,]~‘i’ +(W,-ixy,5i)[6xrk~5j+~iA~jykr,]E’j’ + 3im[i(A
- iy,B)y,rk,
3im;i(A + iy,B)y,t,}
(5.16)
In Eq. (5.14) we use a stroke 1 for the covariant derivative on the threedimensional surface Z including both the afline and the spin connections. Besides, we have used the expression for the covariant derivative of 1 on ,,Y, 9Al,=
[Djl+i(nbLak-n”Lbk)Kjkr,,~]
dxj,
(5.17)
where Djl =
aj2 + $$?~,A.
(5.18)
Analogously, by considering the explicit expression for j given in Eq. (5.4~) of Ref. [ 11, we compute # d3x and 2 can be written as
(5.19) where J=g”2{3X(UI+iy5WI)+~Xrjk[iy5(Ui-X5i)-(Wi-i~y55i]&iik +~i[iyk[(AWj-BUj)-iiX(y,A+iB)~j]Euk+~Ak~iyj~iik -~irr~[~(A-iy,B)y,f~~~“~+3itn~(A+iy,B)y’}.
(5.20)
It should be noted that in all the above expressions both the contorsion tensor and the extrinsic curvature were replaced in the theory by their explicit expressions (in favor of the bilinear function in the spinor fields and the gravitational momenta, respectively).
224
FOUSSATSAND
ZANDRON
The first class constraints Sub, XL, Zk, and S@obtained above generate all the gauge symmetries of the Hamiltonian. Thus the Hamiltonian (5.3) decomposed in this set of first class constraints constitutes the starting point to determine the commutation relations between such constraints. By straightforward but very tedious computation, which we omit here, it is possible to show that the set of constraints L?&(X) closes the constraint algebra, in complete analogy with what happened in the simple supergravity case [3, 133, that is: (5.21)
C-%4(x),xY(Y)1=~=,4B&(y)8(x-y)),
where ACAB = RCA, - Cc,, for curvatures RCA, and Poincare structure constants CM. For the supergravity sector of the coupled system under consideration, it is possible to confront our results with those obtained in [3, 133. We can see that the same terms appear in the corresponding constraints. In particular, our expression for the fermionic constraint 2, in the pure supergravity piece, is in agreement with that obtained in Eq. (14~) of Ref. [3] in which the 3-gravitino term appears. This term does not appear in the expression computed by Pilati in Ref. [13]. On the other hand, from the comparison between our results and those of Ref. [3], we can see that there are some differences of factors in the final expressions. The reasons for such differences are as follows: first, our Hamiltonian differs by an overall factor from that used in [3]; second, we work with the gamma matrices r,, = 2oab and not with ~~~~ and finally, our definitions of the 2-form Ci, the 3-form Q x , and the momentum # (Eq. (4.22)), also differ by factors from the analogous definitions given in [3].
6. DIRAC
BRACKETS
As it was already commented, in the first order CCF all the primary constraints (2.1) are of second class. It is well known [9, 10, 241 that the second class constraints cannot be interpreted as symmetry generators hence they must be eliminated from the theory. This is done by using the Dirac brackets which are obtained from the second class constraints 0, by means of the definition
CF,Gl* = [F, Gl-
[F, 52,1C="[sZ,,,
G],
where C”‘[Q,,, Q,] = dzB. Here the indices C, LI, 6’ are compound all the possible second class constraints. The important properties of the Dirac brackets are: (i)
(6.1)
and Q, denotes
If one of the functions F or G is first class, then
[F,G]*z[F,Gl.
(6.2)
SECOND
In particular
ORDER
for the Hamiltonian
CANONICAL
225
FORMALISM
J? we have
[F, 2’3*
z [F, 21.
(6.3)
This means that the same equations of motion are obtained by using the Poisson or the Dirac brackets. Thus the rate of change in time of any functional F of the canonical variables is also given by P= [F, 2-J*. (ii)
For any functional
(6.4)
F of the canonical variables it is [O,, F]* =O.
(6.5)
Therefore we can set Q, = 0 either before or after evaluating the Dirac brackets. To compute the Dirac brackets (6.1) we must first obtain the set QL of second class constraints by considering the restriction to C of all the constraints (2.1). The second class constraint (2.la) is easily eliminated in the second order formalism. Furthermore, in this formalism Eq. (2.1 b) which contains the spin connection mab and hence the time derivative of the vierbien, is.no longer a constraint. When we take such an equation strongly equal to zero, it allows us to write the expression for the momenta rc,. Thus the set of second class constraints are given by
Q,(x) = [Q,(x), ...?Q,(x)1 = (x*Qz, x*@?,x*‘v,, x*ya,x*@,,4), x *@(B,?x*‘Pa2X*?n)
(6.6)
where Q,, = 7T,-4&y’&@
g’12
(6.7a)
azk
g”’
(6.7b)
= lTk - 4y,y’5’Eiik
Q3 = 6- 6iXy, g”’
(6.7~)
Q, = 0 + 6iy,lg1’2
(6.7d)
52, = IZ,,, + 6( U, + xTii[;
+ a iB[jyk~jsuk)
Q2, = IT{*, + 6( W, + $Xrjkti~iik
g’/*
- $iAryk(jeiik)
(6.7e) g”’
I
(6.7f)
Q,, = YJX)
(6.39
Q,u = E,(X).
(6.7h)
Using Eqs. (3.3), the Poisson brackets between the constraints computed and they are given by
(6.7) can be
226
FOUSSATS
AND
ZANDRON
CQ~i(xhQ*j(Y)I = -8g1’2y,ykEij/c83(X, .Y) [Q,i(x),
Q,(y)]
= -3ig”2(~y,r’k-B~‘~k)&ijkS3(X,
[Q,,(X),
Q,(y)]
=3g”2(XT’k-iAgiyk)&ijkS3(X,
[Q,(x),
Q,(y)]
=
[Q,(x),
l22,(y) -J = 3g1’2[ir,k&~k63(x,
CQ,(X)?
Q,(Y)1= 1w’2Y.~3b, y)
[Q,(x), G,(y)]
-6g”‘[J-b3(x,
(6.8a) y) J’)
(6.8b) (6.8c)
y)
(6.8d)
y)
(6.8e)
= 3ig1’2[jy&ijkd3(x,
(6.8f) y)
[Q&J, %a(~)1= CQ,tx),Q,,b)l = 6g”*d3k
t6.W
Y)
(6.8h) Poisson
and the corresponding brackets for the adjoint Dirac spinor. All the 0th brackets vanish. From the matrix elements (6.8) and by straightforward calculation the elements of the inverse matrix P” can be computed and we find cf = c; = &g-‘12y,yjyi (6.9a) c34 = c43 = & g - 1/2yI (6.9b) cy= -(-;5=cf+ -Cy=&‘wn, (6.9~) cy=
-qy=c;l,‘=
CA;= -c;;=c;;= c;l=
- pJI =
czg= -cd”= Cfi=
c;3
-C~:=~g-“*n,(y,n+By5~i)
(6.9d)
-Cff=
(6.9e)
= _
C;4
-~g-1’*n,y,(y,n+iA5i) = & g- Wn,yiti
cj13= -Cjf4=
(6.9f)
-&g-l12n,y,yi~i
(6%) (6.9h)
-C~~=$g-1i2n,n,[aX(B~,-iA)T,S,+~iykS,le~k.
Using the definition of the Dirac brackets (6.1) and the result (6.9) we arrive at the field-field Dirac brackets, (6.10a) lIlitxL
u~(Y)l*
=
cA7(x)63(x3
Y)
[5itx)3 wa(Y)l* = cAf(x)s3(x~Y) CG), w)l* = -c43w3(x, Y) C4xL u,(Y)l* = cw~3(x~ Y) [4x), wa(Y)l* = c3xv3(x~ Y) CA(x),gal* = C%) d3(x,Y) [B(x), wab)l* = C:*W3(x, v) CU,b), ~bb)l* = C~W3(X> Y).
(6.10b) (6.10~) (6.10d) (6.1Oe) (6.10f)
(6.W) (6.10h) (6.1Oi)
227
SECONDORDERCANONICALFORMALISM
Analogously
the momentum-field
Dirac brackets are
Cn,‘Cx),Lbj(Y)l* = Cn,‘tx), Lbj(Y)l [n,‘(x), p(y)]* = ~&JJ’rkiJ3(x, y) L-n,‘(-~h ay11* = -gwd [n,‘(x), A(Y = 0 Cn,‘(x), Q)l* [II,‘(x),
[mu”,
(6.11b) (6.11~)
+ n,yiy,)d3(x, y)
(6.11d) (6.1 le)
=0
Ub(y)]*=
wb(y)]*
(6.1 la)
-nb(U,L,‘-n,U’-i;lrak5,&lik
=
+
+iB
-n”(
&Lo’-
-
(6.1 If)
y) ncr w’+
$iA[jiya~k&ijk)63(x,
Finally the Dirac bracket between the momentum
Xr,,l,&‘ik
(6.1&J
y).
components
is
(6.12) The Dirac brackets (6.10a, d), (6.11b, c), and (6.12) can be confronted with those obtained in Refs. [ 13, lo] for pure supergravity and for gravity coupled to the spin i field.
7. CONCLUSIONS
We have shown that both formalisms, the CCF and the CVF can be related. This relation is non-trivial and it is obtained by remarking that the form bracket detinition introduced in the CCF and the Poisson brackets used in the CVF may be connected through an integral relationship. For the particular case of the W-Z@ SUGRA coupled system the method to obtain the second order CVF starting from the first order CCF was explicitly developed. We found the expression for the spin connection and we showed that the contorsion tensor depends on the fermionic fields of the model. Moreover, starting from the total Hamiltonian provided by the CCF containing the set of second-class primary constraints, it was possible to arrive at the Hamiltonian as generator of time evolutions, in the CVF. By following this procedure it is possible to show that 2 is weakly zero because it is given by pieces, each one of them is precisely one of the inhomogeneous equations of motion for the supergravitational field with source, plus terms containing constraints. Subsequently, from this expression and by straightforward calculation the Hamiltonian was written in terms of the first-class constraints which closes the supersymmetric
228
FOUSSATS
AND
ZANDRON
constraint algebra. In particular, the piece Y&, = M,,xO (generators of the local Lorentz group) was obtained in a natural way. On the other hand, the second-class primary constraints obtained in the CCF were analyzed. We have shown that this analysis performed in the framework of the CCF is simpler than that given by starting directly from the canonical vierbein formalism. We also found the Dirac brackets based on the second class constraints for all the fields which take place in the model. For the pure supergravity sector (without W-Z fields) the results obtained in this work can be confronted with others previously obtained and the agreements can be established. The above results show how the CCF provides a useful guidance in writing the CVF in a more simple and compact way.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
23. 24.
A. A. A. J. A. A. A.
FOUSSATS AND 0. ZANDRON, Ann’Phys. (N.Y.) 189 (1989), 174. D’ADDA, J. E. NELSON, AND T. REGGE, Ann. Phys. (N.Y.) 165 (1985), 384. LERDA, J. E. NELSON, AND T. REGGE, Phys. Left. B 161 (1985), 294, 297. E. NELSON AND T. REGGE, Ann. Phys. (N. Y.) 166 (1986), 234. LERDA, J. E. NELSON, AND T. REGGE, 1112. J. Mod. Phys. A 2 (1987) 1843. FOUSSATS AND 0. ZANDRON, Classical Quantum Gravity 5 (1988), 605. FOU~~ATS AND 0. ZANDRON, Classical Quantum Gravity 5 (1988), 1231. A. FOUSSATS AND 0. ZANDRON, ht. J. Mod. Phys. A 5 (1990), 725. P. A. M. DIRAC, “Recent Developments in General Relativity,” Pergamon, New York, 1962. J. E. NELSON AND C. TEITELBOIM, Phys. Left. B 69 (1977), 81; Ann. Phys. (N.Y.) 116 (1978), 1. S. DESER AND C. J. ISHAM, Phys. Reo. D 14 (1976), 2505. S. DESER, J. H. KAY, AND K. S. STELLE, Phys. Rev. D 16 (1977), 2448. M. PILATI, Nucl. Phys. B 132 (1978), 138. L. CASTELLANI, P. VAN NIEUWENHUIZEN, AND M. PILATI, Phys. Rev. D 26 (1982) 352. R. ARNOWITT, S. DESER. AND C. W. MISNER, “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), Wiley, New York, 1962. P. A. M. DIRAC, Proc. R. Sot. London A 246 (1958), 333. A. FOU~~ATS AND 0. ZANDRON, Ann. Phys. (N.Y.) 191 (1989), 312. A. FOUSSATS AND 0. ZANDRON, Classical Quantum Graui?y 6 (1989), 1165. A. FOU~~ATS AND 0. ZANDRON, hf. J. Mod. Phys. A 5 (1990), 1671. R. DE AZEREDO CAMPOS AND J. M. L. FISCH, Inr. J. Mod. Phys. A 3 (1988), 2371. A. FOUSATS AND 0. ZANDRON, “Second order Hamiltonian formalism from the CCF. Super Yang-Mills coupled supergravity system,” submitted for publication. Y. NE’EMAN AND T. REGGE, Phys. Let. B 14 (1978) 31; R. D’AURIA, P. Ft&, AND T. REGGE, Riv. Nuovo Cimento 3 (1980), 12; R. D’AURIA, P. FRB, E. MAINA, AND T. REGGE, Ann Phys. (N.Y.) 135 (1981), 237; Ann. Phys. (N.Y.) 139 (1982), 93; L. CASTELLANI, R. D’AURIA, AND P. F&, in “Proceeding 19th Winter School and Workshop of Theoretical Physics, Karpacz, Poland” (B. Milewski, E.), World Scientific, Singapore, 1983. A. SCHOUTEN, “Ricci Calculus,” Springer-Verlag, Berlin 1954. K. SUNDERMEYER, “Constrained Dynamics,” Springer-Verlag, New York, 1982.
OF PHYSICS
203, 207-228 (1990)
Matter Supermultiplet Coupled to Supergravity in the Second Order Canonical Formalism A. FOUSSATS Facultad U.N.R.
AND
0. ZANDRON*
de Ciencias Exactas Ingenieria y Agrimensura, Av., Pellegrini 250, 2000 Rosario, Argentina
Received December 27. 1989
Starting from the first order canonical covariant formalism (CCF) for the Wesss Zumino @ SUGRA coupled system, we find the second order canonical vierbein formalism (CVF). The primary constraints obtained in the CCF, are analyzed in the framework of the second order CVF. From the total Hamiltonian density obtained in the CCF and by making the space-time decomposition, the Hamiltonian in the CVF is found. This is the correct Hamiltonian which generates the time evolution of generic functional of fields and momenta. It remains defined by means of the set of the first class constraints of the system. This set closes the well-known constraint algebra. #n 1990 Academic Press, Inc.
1.
INTRODUCTION
In a previous paper [l], the canonical covariant formalism (CCF) on group manifold for the supersymmetric Wess-Zumino matter multiplet coupled to supergravity (W-Z) 0 SUGRA was constructed. As it was already shown in several works [2-81, the first order CCF is particularly useful for treating systems which are gauge invariant under some group of transformations. When the CCF is applied to such systems the general result obtained is that all the constraints are primary (there are no secondary constraints in the formalism) and none of them is first class. This relevant characteristic together with the covariance of the formalism in all their steps, allows us to obtain the constraints, the total Hamiltonian, and the consistency condition for the constraints in a very simple way. Thus the equations of motion and the rheonomic conditions are obtained by means of straightforward algebraic manipulations. In particular, the correct Hamiltonian as a first class dynamical quantity remains given unambiguously once the primary constraints are computed from the Lagrangian density. The total Hamiltonian thus defined is strongly conserved, i.e., dH=O.
On the other hand, it is well known that the Hamiltonian description of the gravitational field by means of vierbein and spin connection [g-14] is more * Fellows of the Consejo National de Investigaciones Cientilicas y Tecnicas, Argentina.
207 0003-4916190 $7.50 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved
208
FOUSSATS
AND
ZANDRON
adequate than that given through the metric tensor and the metric affine connection [ 15, 161. In the framework of the canonical vie&in formulation (CVF) for gravity and supergravity, both the first [ 141 and the second [lo] order formalisms must also be distinguished. In the second order formalism for pure supergravity, the spin connection is a dependent functional of the vierbein and gravitino fields. The construction of the second order formalism starting directly from the first order CVF involves very heavy algebraic manipulations [13, 141. These difficulties are even enough, when a supersymmetric system (i.e., Yang-Mills or Wess-Zumino supermultiplets) coupled to supergravity is considered. Moreover, it is well known that the transition from the first order formalism to the second order one, allows us to separate the true dynamical degrees of freedom from those of gauge degrees of freedom. Thus the importance of treating these constrained systems in the second order Hamiltonian formalism, grows up when the quantization of such systems is considered. Also this kind of formulation should be useful for implementing new ideas about the quantization of the gravitational field. At this stage the CCF plays an important role, because of its more simple and compact structure. In particular the CCF can be used as an interesting formal mechanism to derive and analyze the set of constraints in CVF. Really, the CCF does not define a standard mechanical system in the sense that it is not a proper Hamiltonian theory as it is in the case of the CVF which propagates data defined on an initial hypersurface C [9]. Consequently, the first order CCF and the CVF must be related, and it is possible to show that this relation is non-trivial [4, 71. The first question is that in the construction of the canonical covariant formalism, the form-brackets are introduced and they must be related to the usual Poisson brackets defined in the CVF. Besides, it must be pointed out that the CCF takes the exterior derivative d as a form-observable and it has no direct analogue in the CVF. Thus, in the CCF the first class dynamical quantity defined as the Hamiltonian density, is not the Hamiltonian which generates the time evolution of generic functionals of fields and momenta. Therefore, starting from the bosonic 4-form HT defined in [ 11, we must construct the generator of the time evolution. This must be done by finding the set of first class constraints which are the generators of the constraint algebra [lo] and the Hamiltonian remains given as a functional of these generators. As we will see, in practice the CVF can be recovered from the CCF by taking into this last formalism some field equations of motion as constraints. For simple supergravity, the first order CVF was directly constructed [13, 14) and it was also found starting from the CCF [4]. Both results were confronted and the agreement and differences were pointed out. Recently [21], starting from the CCF we have constructed the second order CVF for the super Yang-Mills theory coupled to supergravity in dimension D = 4. In this paper we have treated the W-Z@ SUGRA coupled system, because it constitutes an interesting example for the application of the CCF on group manifold. In this system the first order formalism, requires the introduction of two kind of fields. The geometrical l-form gauge fields to describe the supergravity
209
SECONDORDER CANONICALFORMALISM
sector and a set of non-geometrical O-form fields for the description of the Wess-Zumino matter supermultiplet. Moreover, this coupled system allows us to introduce in the CCF the definition of supercurrent of the W-Z matter multiplet which acts as a source for the supergravitational field. It also allows us to obtain information about the set of supergravity auxiliary fields. As in this system the torsion equation is different from zero, this equaton gives rise to a modified expression for the Lorentz spin connection. As we will see later on, the spin connection obtained from the torsion equation is a functional of the vierbein, the gravitino, and the Majorana spinor O-form field 2, partner of the boson fields A, B which are necessary to describe the W-Z supermultiplet. Finally, we comment that the purpose of this kind of treatment is to give a second order component formalism for non-linear supergravities in the Riemann curvature, starting from the CCF on group manifold previously developed by us (see Refs. [17-191). At least for the linear case it should be important to confront the results obtained from the CCF with those given in Ref. [20], for the interesting SYM @ SUGRA N= 1 D = 10 coupled system [ 173. The paper is organized as follows: In Section 2, we briefly summarize the definitions used and the results obtained in the construction of the CCF for the W-Z @ SUGRA coupled system. In Section 3, we make the space-time decomposition in M4 = G/H and we give the relation between the form-brackets and the Poisson brackets. In Section 4, starting from the CCF we construct the second order CVF by writing in components all the quantities which in the CCF are written in form language. The general expressions of the formalism restricted to the three-dimensional surface ,E are found, and the primary constraints obtained in the CCF are analyzed in the framework of the second order CVF. In Section 5, we explicitly compute the set of first class constraints which verify the constraint algebra. Using these constraints, the Hamiltonian which generates the time evolution of a generic functional of the fields and momenta remains defined. Finally in Section 6, we analyze how the Dirac brackets for the system under consideration can be obtained.
2. THE CCF FOR W-Z@SUGRA
SYSTEM: DEFINITIONS
AND RESULTS
We summarize the relevant results obained in Ref. [l] in the construction of the lirst order CCF on a group manifold G, for W-Z@ SUGRA constrained coupled system. The dynamics is described by introducing the set of pseudo-connection l-forms pA = (c@, V”, 5”) and a set of three O-forms. The set of l-forms are the gauge SUGRA fields and the three O-form fields are necessary to describe the supersymmetric W-Z multiplet (0, 0, 4). These O-forms dynamical fields are A, B (bosonic objects), and A (anticommuting object), transforming respectively as a scalar, a pseudo-scalar, and a Majorana spinor. The invariance under global supersymmetric transformation laws and the rheonomic symmetry conditions in a first order formalism require the introduction of two O-form independent dynamical fields U”
210
FOUSSATS
AND
ZANDRON
and W“ transforming respectively as a vector and axial vector under the Lorentz group. Therefore, the first order CCF for this coupled system is constructed by starting from the set of eight dynamical fields namely: cPb, Vu, <“, A, B, A, U”, W“ and we will denote their corresponding canonical conjugate momenta by ‘~t,~, x,, 71x0 7q‘4), QY), $3 z,, Ea. The Lagrangian density for the system allows us to obtain the eight primary constraints given by @ab=71,bM0
(2.la)
@a =lt n -22WbCA Vd&abed -0 -
(2.lb)
@” = 710L - 4(y,y,5)”
(2.lc)
@(A)=
I’+,+
uavb
+$iB([r, @(B,=~C(B,+
A V” z50 h
~“5) wavb
A
-@([Ay”<)
v’ A
V,zO
v’
A
A
y=$+iyUnVb
vd&,bcd-6&‘~~ob~
A
A V’A
A
vu
A
vb
(2.ld)
I/dE,bcd-
6i/iz,bc
A T/”
A
vb
(2.le)
v,E:o
Vd&abrdzO
(2.lf)
(Pa=ToXO
Wg) (2.lh)
q, = E, Fz0.
From the canonical Hamiltonian given in Ref. [ 1, Eq. (3.4)] and the constraints (2.1), we can define the total massive Hamiltonian as ff,(WZ) = H,,,(m) +z’
+ /tub A @jab+ A0 A y’,+c”
A Cpa+a”
A 0,
+/I”
A @a + ~4~~)
A @CA1
+ /itB)
A Q(B)
A va,
(2.2)
where H,,,(m)
= H;,U,GR* + H~,Z(O) + H,WZ.
(2.3)
The Hamiltonian pieces @zGRA, Hz:(O), and HF were explicitly written in Eqs. (3.5), (3.6), and (3.10) of Ref. [l], respectively. We have shown, that the total Hamiltonian (2.2) is the correct first class dynamical quantity which describes the dynamics of the constrained coupled system under consideration. Moreover, we remember that in the CCF all the constraints are primary (there are no secondary constraints in the formalism) and none of them is first class (see Ref. [l, Eq. (4.1)]. When the fundamental equation &I = (A, HT) + aA, involving the form-brackets for a generic form A is used for the constraints, obtain the following results:
(2.4) we
SECOND
(i)
For the compound
ORDER
CANONICAL
FORMALISM
211
index A = (ab, a, c(), the equations, d@,=(@,,
H,)zO.
(2.5)
These are the equation of motion for the supergravity sector (see Ref. [ 1, Eqs. (4.6)]. In the inhomogeneous supergravitational field equations (4.6) of Ref. [l], the quantity J= (Jab, J,, J,) 3-form multiplet appears which is the Osp(4/1) supercurrent of the W-Z matter multiplet acting as source for the supergravitational field. In the framework of the CCF they remain defined by
Jab= (@uh,H,W=)
(2.6a)
J, = (@,, H,W=)
(2.6b)
25= (cs, fly=)
(2.6~)
and they can be written as a function of the set of non-minimal auxiliary fields for supergravity (A,, Ai, Y, 9, x) and the spinor-vector j, as we have shown in [l]. In particular, the axial vector A, is defined by A, = 32y5 y,ll
and therefore the component
(2.7)
Jab of the supercurrent can be written as Jab=;A,V”
A V, A V,,.
(2.8 1
From the torsion equation d@ab = (@ab, HT) ~00,
(2.9)
we obtain R”=
-LPbcdAbV~ 8
A V,.
(2.10)
(ii) For the compound index M which takes into account the five O-forms dynamical fields, the equations d@,=(@,,H,)=O,
(2.11)
give rise to the five Eqs. (4.3) of Ref. [l] for the W-Z sector in the massive case. Those equations determine the dynamics of the W-Z system and from them the following rheonomic conditions for the fields A, B, and A.can be obtained: dA=
U,V”-;5~
dB= W,V”-i;iy,<
Gal= x, vu - i( iy, U” + ysya wa) 5.
(2.12a) (2.12b) (2.12c)
Equations (2.12) show the rheonomic symmetry of the total Hamiltonian (2.2) and they also make it possible to turn the first order Hamiltonian formalism into
212
FOUSSATS
AND
ZANDRON
the second order one. Moreover, from Eqs. (2.12) we can see that the O-forms bosonic fields U” and W” introduced above and the Majorana spinor-vector field X, represent the spatial derivative of the W-Z fields A, B, and I, respectively. All other results about CCF that we wll use in what follows are given in Ref. [ 11. 3. THE SPACE-TIME
DECOMPOSITION AND THE RELATION BETWEEN POISSON AND FORM BRACKETS
With the purpose of constructing the second order Hamiltonian formalism for the system under consideration, we will follow a similar procedure to that given in [3,4] for gravity and pure supergravity. We use also our previous behaviours, notations, and conventions given in the paper [21]. The first step is to carry out the space-time decomposition. Subsequently, all the equations and quantities which we have been writing in form language using exterior calculus on differential forms, must be written in components. We will use for the Greek and Latin indices the same convention we have used in [21, Section 2.23. All the dynamical fields must be considered only as reduced forms, that is, they are forms defined on the coset manifold M4 = G/H (physical superspace), where H is a subgroup of the Lie supergroup G and it is the exact gauge symmetry group. Afterward, the extended forms can be obtained naturally for the whole supergroup G, through H gauge transformations [22]. Moreover, we consider that the reduced forms defined on M4 are written in the holonomic basis dxP. Finally, equations, fields and forms must be projected on a space-like x0 = t = to hypersurface 2 of three dimensions. This requires the introduction of the injection map x: C -+ M4, in such a way that the associated pull-back x* acts on any generic forms by setting t = to and dt = 0. For the space-time decomposition in M4 we follow Ref. [15]. The vierbein V” = Lf dx” is split according to the expression (2.11) of Ref. [21]. Besides, we use, the notation given in Eqs. (2.12t(2.15) and the properties given in Appendix C of Ref. [4] for the alternating tensor sUbrd in tangent space and the alternating tensor siik on ,X. We use the above mentioned expressions of Refs. [21,4] whenever they are necessary. We call p;(x) = [ --~II(~‘;~ (x), L:(x),
- “*Tc’,(x)E~~ dx’ A dxk = g- %;(x)&,
where the 2-form zi is defined on the hypersurface C.
(3.1)
SECOND
ORDER
CANONICAL
213
FORMALISM
Analogously, for the other O-form dynamical fields CL””= (A, B, 1, U”, IV’) the corresponding canonical conjugate momenta rr,,,,= (rccA,,Q), 8, r,, E,) are 3-forms, therefore we have the definitions, 71M=sg
1
-“‘I~,(x)E~~
dxi A dx’ A dxk = ZZ,(x)Q,
= n,,,,(x) d3x, (3.2)
g-“2
where the volume 3-form Q, is defined on Z and n,(x) = (n,,,(x), Z~,,,(X), e(x), K(X)? &b)). For the constraints (2.1), similar expressions as those given in (3.1) and (3.2) for the momenta hold. Then, the usual Poisson brackets between the components of the fields and canonical conjugate momenta are given by [L”;(x),
74(y)]
= - [Q’(y),
L”;(x)]
= 6”&J3(x,
y)
Cmiab(x),~jd(Y)I = - C~(‘,db), oi”“(x)l = Sab[cd,cw3(X,y) Cti(X)9
n/?j(Y)l
=
Cna’(Y),
gZi(x)]
=6”/j6/63(X,
y)
(3.3a) (3.3b) (3.3c)
cm)? ~,(.Y)l = C&(Y), ax)1 = QB3(x, y)
(3.3d)
[uu(x), 6,(y)] = - [%(v), vu(x)] = b”b~3(-%y) CA(x), n,,,(Y)1 = - L-n,,,(Y)? A(x)1 =63(x, Y)
(3.3e)
[ w”(xh
Eb(Y)l
=
-
[&b(Y),
w”(x)l
=
&?j3(x,
ev)
[B(x), n,,,(y)1 = - [n,,,(y), B(x)1 = J3(x, Y).
(3.3f) Wg) (3.3h)
The Dirac delta function appearing in the above equations is defined by
b3(x, ev)f(x) =fb), sd3x
(3.4)
without appeal to the metric. As was briefly commented in the introduction there is another question to take into account to relate both formalisms the CCF and the CVF. In the CCF the form-brackets are defined and they play an important role in the construction of the Hamiltonian equations of motion and in the determination of the properties of the constraints. Such form-brackets must be related to the Poisson brackets defined in CVF. As was shown in [4] the Poisson brackets yields more information than the form brackets. Moreover, a particular subset of Poisson brackets can be related to the form-brackets by means of an integral relationship. Taking into account the properties for the form-brackets given in Eqs. (2.3) and (2.4) of Ref. [l], we are going to write for supergravity an analogue integral relationship to that given in [4] for gravity. We assume that the generic forms A(x) and B(y) at the points x and y, respec-
214
FOUSSATS
AND
ZANDRON
tively, on the hypersurface C, are polynomials in the fields and momenta variables only. The forms A(x) and B(y) have a and b degrees and Fermi grading IA 1 and IBl, respectively. The Poisson brackets between forms [A(x), B(y)] define a form of degree a+ b on Cx Z. Because of the way they were defined, the Poisson brackets between forms do not contain derivatives of the Dirac distribution and they can be evaluated by expanding the forms A(x) and B(y) in the holonomic bases dx’, dy’ and then using the Poisson brackets (3.3) between fields and momenta components. Thus we can write the following Poisson brackets between dynamical canonical conjugate (forms) variables:
CP”(X),~Ay)l=;
hA, g-1’2(~)~&) hi * &’ * dykd3(x, Y)
=P,g-l’*(y)dxj
A C,(Y)~~(X,
y)
(3Sa)
and b”(x),
n,(Y)1
1 =- 3! d”Ng - “*( Y)E~~( y) dy’ A dy’ A dyk d3(x, y) = P,
g-1’*(y)i2,63(x,
(3.5b)
y).
The Poisson brackets between generic forms A, B, C of degrees a, b, c and Fermi grading IA 1, 1B(, and ICl, respectively, remain defined by means of the definition (3.5) and they verify the following properties: [A(x),
B(y)]
= (- l)“b+ 1+‘A”E’ [B(y),
A(x)]
(3.6a)
CA(x), B(Y) * C(Y)] = [4x), B(Y)] * C(Y) +(-l)“b+‘A”B’B(y) A [A(x), C(y)]
(3.6b)
CA(x) * B(x), C(y)1 = A(x) * L-W), C(Y)] + (- l)b’+‘B”C’ [A(x), C(y)] A B(x)
(3.6~)
(- l)“‘+‘“““[A(x), +(-I)
[B(y),
C(z)]]
cb+‘C”e’ CC(z), [A(x), B(y)11
(- l)b”+ ‘We’ [B(y),
[C(z),
A(x)]]
= 0.
(3.6d)
Now, the integral relationship which allows us to relate the form-backets subset of Poisson brackets between forms is given by
with a
A CA(x), B(Y)] A KY),
(3.7)
t-11 a+l+‘A”B’jzor(x)
A (A, B) A /3(x)=/jzXZa(x)
SECOND
ORDER
CANONICAL
215
FORMALISM
where CI and/I are two test forms of degrees 3 --a and 3 - 6, respectively, and arbitrary Fermi grading. We can check (3.7) for pairs of canonical variables. If we consider that the form-brackets for pairs of canonical variables is [ 1 ] (/iA, 7c8)= (-l)o+‘+‘A’8AB,
(3.8)
the left hand side of (3.7) is equal to LiA,J, LX(X) A b(x). Taking (3.5) for the Poisson brackets, the right hand side of (3.7) is
the expressions
4x) A CPA(X)3 7-c&)1 A P(Y) 1.j zxz a(x) A dx’ A dy’ A dyk A b,(y)
=
hA,
jz
a(x)
A
dy'g-1'2(y)&,~(~)~3(X,y)
dx’ J: sz,6!B,(Y)g-“2(Y)~3(x,
=dA, s,dx)Adx’s, Ph4S3(x, Y) d3y
= dA,
Y)
j
4x1
z
A P(x).
(3.9)
This last result and the property (3.6a) of commutation for the Poisson brackets, shows that the form-bracket definition and the definition of the Poisson brackets for the components of the canonical conjugate variables are consistent.
4. THE SECOND ORDER CVF FROM THE CCF
To find explicitly the second order canonical component considering the following expression for the torsion R”=dV“-wab
A
i’-,-$[A
formalism,
we begin by (4.1)
~“5,
associated to the graded Poincare algebra and the solution (2.10) for the torsion provided by the torsion field equation (2.9). From these equations it is possible to obtain the solution for the l-form spin connection gab of the W-Z@ SUGRA system, whose components in the holonomic basis are generally given by G/y
v, 5, A) = zY/b( V) - qb(&
A).
(4.2)
(4)
Here we write hrab( V) t o indicate that such a functional is the same functional of the vierbein as in pure gravity. The second piece of the right hand side of (4.2) is the component of the contorsion tensor C, which is a functional of the gravitino field 5 and the axial vector A, defined in (2.7). To find the explicit expression for the contorsion tensor we proceed as follows. 595/203/l-15
216
FOUSSATS
AND
ZANDRON
As it is well known the full covariant derivative including world (metric) connection satisfies the metricity condition
both the spin and the
i?,L", + (FipL,, -v-plrvLap = 0,
(4.3)
where 4P’ TVis the affme connection of the four-dimensional manifold M4 and is no longer symmetric in its lower indices because of the torsion generated by the spinor fields 5 and 1. On the other hand, by considering the spin connection (:)Ob and the affine connection 3r:k (no longer symmetric in his lower indices) on the three surface L’, it can also be written as
akLq + 'A>L,-
3rkj~ai = 0,
(4.4)
by the vierbein postulate on the three-dimensional surface Z. Moreover, by multiplying Eq. (4.4) by the normal n, to the surface C we obtain the equation
d,n"+'i>n,=O.
(4.5)
Now, we consider the restriction to the three-dimensional surface Z of the torsion expression (4.1) rewritten in components and the explicit expression (2.10) for the torsion R". Thus we obtain
a,,Lq, + !2$ L,, - t[ckyyjl Taking
+ ~~~~~~~~~~~~~ = 0.
(4.6)
into account Eq. (4.3) and Eq.(4.6) we can find
sp, E g4rpkj- “rpjk) = f[,,yp
(4.7)
We note that in the second orderC4FVF EqQ4.3) and (4.4) allow us to determine completely both spin connections mpab and mkab. After some algebraic manipulations, the following relationship between the components of both connections can be found: (ziab = (:)pb + (nbLaj - naLbj)Kii. In (4.8) K, is the extrinsic curvature of the three-dimensional manifold M4 and it is defined by
Ku=
~(-g,,+N,,,j+N,,I,)-cji,.
(4.8) surface C in the
(4.9)
The double stroke 11in (4.9) denotes the covariant derivative on the three-surface C, only including the aftine connection.
SECOND
The contorsion
ORDER
CANONICAL
217
FORMALISM
tensor C is defined by (4.10)
CPjl” = - W,,” + svpp + S,,“) and the component
C,, is given by Co, = -f(giYj5,-riY15j+5jYi51)-~&iikAk
(4.11a)
Ciik=-~(giYj5k-giYk4j+~jYiSk)+$EijkAi.
(4.1 lb)
From Eq. (4.8) we can see that the information provided by Eqs. (4.3) and (4.4) are quite different. As the affrne connection 3ri(jkJ in (4.4) is a functional of the metric ‘gY on C and therefore of the Lai only, then (%)kabresults as a functional of the Loi and its spatial derivative only. While, from Eq. (4.3) the restriction to the three-dimensional surface z of the affrne connection 4Ti(jk) and the spin connection (4)
&kub on M4 are functionals of both the components Lai of the vierbein and the corresponding canonical conjugate momenta through the extrinsic curvature K,,, as we will see later on. Of course, the complete expression for the two spin connections involved in (4.8) also differ in their respective parts containing the contorsion. We note that Eqs. (4.8) and (4.9) are formally identical to the corresponding equations given in Ref. [3] for the pure simple supergravity. As we can see from (4.11) in the W-Z @ SUGRA coupled system, the difference is contained in the components C,, and C, of the contorsion tensor C. Such components are not only bilinear fuctionals of the gravitino field 5 as in pure supergravity, but they also contain the spinor field A of the W-Z matter supermultiplet. That is to say, in the system under consideration the contorsion C as well as the torsion S are generated by the two spinor fields which are present in the theory. Now we are going to analyze the primary constraints (2.1) obtained in CCF. We begin by making a brief commentary about the constraints. The possibility of using different Lagrangian densities means that there is not a unique set of canonical conjugate momenta and consequently there is not a unique set of primary constraints in the CCF. For example, if we start from the following Lagrangian density for supergravity dP=Rab
A V’ A Vdc abrd
+ 4p
A Y5Yatf
A
va/“,
(4.12)
which differs from a total exterior derivative of our starting Lagrangian density, we cannot obtain the primary constraints (2.la) and (2.lb) but rather we obtain sob
=
n,b
-
@,=71,x0.
v’
A
Vd&,brd
% 0
(4.13a) (4.13b)
Moreover, by looking at the second order formalism [9, lo] we see that the momentum components dab are strongly equal to zero. Therefore, when both formalisms are related and we go from the CCF to the second order CVF, it is
218
FOUSSATS
AND
ZANDRON
made evident that it was a good choice to start from the Lagrangian density which provides Q, z 0. Another question to take into account is that the CCF plays, with respect to the first order CVF, an analogous role to that played by the CVF of first order with respect to the second order CVF. Thus we will consider that all the primary constraints in the CCF remain at least weakly zero in the CVF. Therefore, in our case we are going to consider that the constraints (2.la), (2.lg), and (2.lh) are strongly equal to zero: Qab = n,b = 0,
(Pn=Ta=O 7
q,=&,=o.
(4.14)
On the other hand, we also assume that the restrictions to C of the constraints (2.lc), (2.ld), (2.le), and (2.lf) are strongly equal to zero, so that the following equations hold: X*w = 0,
x*q,,
= 0,
x*@(B) = 0,
p!P=o.
(4.15)
For the remaining constraint Q,, which is a functional of the spin connection we consider the restriction to Z and take it as a weakly zero quantity, that is, x*cDa E *o x 0.
(4.16)
In the second order CVF, (4.15) correspond to second class constraints, which can be eliminated from the theory by defining the modified Dirac brackets. As we will see below these new Dirac brackets can be evaluated by starting from the second class constraints. Here Eqs. (4.15) allow us to solve for the corresponding momenta K,, 7ccaj, rccBj, and 9,. Equations (4.14~(4.16) among others, show that all weakly zero quantities in the CCF remain at least weakly zero in the CVF as it was commented above. Now, we must compute explicitly $,. Using for the 2-form $. an analogous expression to that given in (3.1), after some algebra we arrive at *a=g-‘/**&=
17,+4(L,kg”-L,‘g’k)Kkj~i+4n,CiikgikCi,
where the explicit expression for the component t/$; = rr,i + 4g’12(Lakgij - Loigjk)Kkj According to the available momentum 2-form 17,:
literature
(4.17)
$=’ is + 4g”*n,C;,
gj”.
(4.18)
[3, 133, we have defined the following
(3)
n, = n, + 2 dfd h vb&,bcd = n, + 2(:)cdA Vb&&cd + 4n, c:k gjkci.
(4.19)
SECOND
ORDER
CANONICAL
219
FORMALISM
(3) We note that the functional which was denoted by bed in (4.19) is the same functional of the vierbein T/” like that of pure gravity without spinor terms. Thus, the Z7, definition is formally identical to that given in Ref. [3]. Again, from (4.19) we can see that the information about t&e Wess-Zumino mater supermultiplet is contained in both, the spin connection gcd on ,X and the components C, of the contorsion tensor C. The importance of Eq. (4.19) is clearly shown because it relates the new momentum 2-form Ii’, with the momentum n, defined in the CCF. From Eq. (4.19) an expression relating the components 17,’ and .i can be easily written. In what follows we will use the momentum Z7, to find the remaining quantities in the second order CVF. Now, it is possible to define the following antisymmetric weakly vanishing quantities Mab : Mobd3x=$,
A V,-$,
A V,zO.
(4.20)
Starting from the CCF, these quantities appear directly as primary constraints because they are functionals of the constraints tin which are the restriction to z of the primary constaints Qa. These six constraints Mnb are the generators of the local Lorentz group for all the fields of the theory and in the CVF they are first class constraints [lo]. Moreover, as we can see from (4.20) they must be considered as densities. Using Eq. (4.18) and performing some algebraic manipulations, the generators Mab can be written as M,, = +(IZ,‘Lbi - IliLa;)
+ 2(n, Lbi - n,Lai) Cik gjkgl/*
-I- 2(L,‘Lbk - L,‘Lak)Cjk,
(4.21)
g’?
Now, we must define other quantities which will be useful later on. If we call # the symmetric tensor density canonical conjugate momentum of the three metric g,, it can be written as .u
E ;(n,iLaj
+ n,jLai)
=
-g1/2(~(f)
_ gtjK),
(4.22)
where ZP’ is the symmetric part of the extrinsic curvature given in (4.9) and K is the trace. These relations involving K” and the momentum rc’i hold, Kc”‘=
-g-1/2(n”-
igFlt)
(4.23a)
2
K= $‘3 ~~K..-K2=g-1(n”x..J’
where Kcyl is the antisymmetric
(4.23b) 0
part of Ku.
ln2)+KCGl~ 2
,_
[Jrlr
(4.23~)
220
FOUSSATS
Before concluding
AND
ZANDRON
this section we give the set of useful relations, M’k
=
LaiLbkM
(4.24a)
nb
Mil
= -L”$,~M,~
(4.24b)
nbnbi = 2~‘~ + 4p’c,‘,
gjk
(4.24~)
Lbinbj = 4,.&i _ MJ _ 4gWcUill which we will use in the explicit computation
5. THE FIRST CLASS CONSTRAINTS
(4.24d)
9
of the first class constraints.
AND THE HAMILTONIAN
Now, we must find the Hamiltonian 2 for the W-ZOSUGRA system, as generator of the time evolution for generic functional of field and momentum variables. The rate of change in time of any functional F of the canonical variables is given by the Poisson bracket of F with this total Hamiltonian, i’= [F, $1.
(5.1)
From the bosonic 4-form (2.2) provided by the CCF, we first make the space-time decomposition of M 4. Next, by choosing the time variable so that the l-form dx” can be detached, we have
[H,(m)=J’dx”A 2 and we consider the remaining bosonic 3-form dimensional surface C. The Hamiltonian 2 turns out to be of the form,
(5.2)
2
integrated
in the three-
x&(X) +L”&%(x) +2(x)&l] d3X, where & are the time components of the l-form field #. To compute explicitly &(x) we consider the expression (2.2) and (2.3). explicit expression for H,,,(m) was given in Ref. [l, Eqs. (3.5), (3.6), (3.10)]. write the piece of HT containing the constraints we use Eqs.(4.14) and (4.15) we also take into account Eqs. (3.8) of Ref. [l] for the Lagrange multipliers. Subsequently, to construct the expression (5.3) we write explicitly the component of all the forms present in HT. After some algebraic manipulations
(5.3)
The To and time and
SECOND
ORDER
CANONICAL
221
FORMALISM
by neglecting total divergences, it is possible to arrive at the following expressions for the 3-forms J&(X) d3x: A?a,d3x=2{(-2RC~
V’E,~,~+J~~)+$[@~A
Sad3x={(-2Rb”/t
Vde,,,.,+4~~/\y,y,~+J,) AVd+ob,]
- hkd %‘d3x=
{(-8LG&y, - ih”
A
cDu>
(54a)
V,]}lzzO
(54b)
A @h}lrzo
V”+4[y5yu
A
T/,-c@~A
A
R”+25) (5.4c)
(,zO.
We note that the expressions enclosed in the parentheses in (5.4a, b, c) are the inhomogeneous equations for the supergravitational field with source (see Ref. [ 11). Thus, Eqs. (5.4) clearly show that .&,, &, and s%?are all weakly zero quantities and they are the first class constraints of the coupled system under consideration. The field equation of motion and the rheonomic equations in the CCF are obtained by imposing the conditions (2.5) and (2.11) on the constraints. These equations are the covariant extension of the preservation of the constraints in time demanded in the CVF. Therefore, starting from the CCF it is possible to see how certain constraints are given directly by the field equations of motion, as was already commented on in the introduction. In the CVF analogous constraints appear when the preservation of the constraints in time is demanded [13]. The results obtained in (5.4) must be expected and for the pure supergravity sector we can compare our results with those previously obtained in Refs [3,4] and we can check the consistency of both results. Now, we must write in components the expressions (5.4) for ZA(x) restricted to the three-dimensional surface C. We begin by computing the 3-form J& d3x. The second order formalism implicates the elimination of the spin connection as an independent variable. That is to say, we must use the explicit expression for the torsion RaIz and solve it for the spin connection. This is equivalent to taking the equation 2R”
A
as strongly equal to zero. Taking
VdEabcd
-
Juh = 0,
into account the definition
&,d3x=1C/,
A
V,-t+bb
A
(5.5)
(4.20) we find
V,s14,,d3xz0.
To compute *U d3x, the initial step is the use of the Gauss-Codacci [23]. Thus the first piece of ZU d3x can be written as 2Rb”
A
VdE,bcd
=
(5.6)
equations
[4n,( ‘R + K2 - K”K,,) + 8Lak( gvKjkili - K,,/,)] g”2 d3X.
(5.7)
222
FOUSSATS
AND
To evaluate the second piece containing
ZANDRON
9[ we use
where
gi[j=aifj-
$$;b[jr*b.
(5.9)
The covariant derivative on the three-surface Z is given by (5.10) therefore the following relation holds: gi
$(nbLak - n”Lbk)Kiktjrab.
(5.11)
To write in components the 3-form J,, we consider for it the explicit expression given in Eq. (5.4b) of Ref. [ 11. That expression contains &I, dB, and 9A. In the second order formalism we must replace these quantities by their rheonomic solutions (2.12). Finally, we must consider the expression (4.18) for the constraints $0’ and by using Eqs. (4.21)-(4.24) after a long and rather heavy algebra it is possible to write Xa in components. Thus, by considering the decomposition LaoXa= N,c%?’ +Nkxk
(5.12a)
L; J, = NI J’ + NkJk
(5.12b)
we arrive at ‘/23R+,g-9,‘+2
P=4(-g -1
128 g
1’2AiAi-
‘712)+~g1’2giy15’5iy15j
&g1/2([jy1(i)AkEijk
+ ;g1’2Di[jysy’~kc’ik} +J’+ajMi’+MU(-~giyItli+~Ak&,jk) zk = gni kllr.+ gg’/zg kJ.(-i[+ 2 ei+ leiilA ) I 8 1 II - 2g”2Dj[iysy,~k& -
2i~kYjri~nii+g1’2(riyj5k)(riYIrj)
+
$e’Y’~k&,,k+
+ +ajMjk+
(5.13)
$(Digk - D,&)x’
Jk +M”($&y,<,+
$iIEuk)
+MiI[g-1’2(nik--gikn)+f~k-,y.5i-~A’&ki,],
(5.14)
SECOND
ORDER
CANONICAL
223
FORMALISM
where J’ =g”*{
-6(U,
U’ + W, W’) + $A’[iyk
+ 6iDjxyjl-
$AkAk
-i(lJi-~~i)[6&,r’k~j-~B[jyl~k]~vk + ( Wi-isXy,<,)[6rITikcj+
iiA[jy,ck]Evk
+ 3m(mA2 +mB* + 212) - 6im&A + iysB)y’<, + 3imti(A Jk=g1’*{6;i(U’
+ iy,B)y,f’k~jEYk} +iys W’)r,+
(5.15) fAk[iy,
$A,[iyk[j~Q’
-6iD,xy.il-6(U,-/&)U,-6(W,-i;zy,t,)W, -i(Ui-@i)[6~y,rk,~j+$B[jyk<,]~‘i’ +(W,-ixy,5i)[6xrk~5j+~iA~jykr,]E’j’ + 3im[i(A
- iy,B)y,rk,
3im;i(A + iy,B)y,t,}
(5.16)
In Eq. (5.14) we use a stroke 1 for the covariant derivative on the threedimensional surface Z including both the afline and the spin connections. Besides, we have used the expression for the covariant derivative of 1 on ,,Y, 9Al,=
[Djl+i(nbLak-n”Lbk)Kjkr,,~]
dxj,
(5.17)
where Djl =
aj2 + $$?~,A.
(5.18)
Analogously, by considering the explicit expression for j given in Eq. (5.4~) of Ref. [ 11, we compute # d3x and 2 can be written as
(5.19) where J=g”2{3X(UI+iy5WI)+~Xrjk[iy5(Ui-X5i)-(Wi-i~y55i]&iik +~i[iyk[(AWj-BUj)-iiX(y,A+iB)~j]Euk+~Ak~iyj~iik -~irr~[~(A-iy,B)y,f~~~“~+3itn~(A+iy,B)y’}.
(5.20)
It should be noted that in all the above expressions both the contorsion tensor and the extrinsic curvature were replaced in the theory by their explicit expressions (in favor of the bilinear function in the spinor fields and the gravitational momenta, respectively).
224
FOUSSATSAND
ZANDRON
The first class constraints Sub, XL, Zk, and S@obtained above generate all the gauge symmetries of the Hamiltonian. Thus the Hamiltonian (5.3) decomposed in this set of first class constraints constitutes the starting point to determine the commutation relations between such constraints. By straightforward but very tedious computation, which we omit here, it is possible to show that the set of constraints L?&(X) closes the constraint algebra, in complete analogy with what happened in the simple supergravity case [3, 133, that is: (5.21)
C-%4(x),xY(Y)1=~=,4B&(y)8(x-y)),
where ACAB = RCA, - Cc,, for curvatures RCA, and Poincare structure constants CM. For the supergravity sector of the coupled system under consideration, it is possible to confront our results with those obtained in [3, 133. We can see that the same terms appear in the corresponding constraints. In particular, our expression for the fermionic constraint 2, in the pure supergravity piece, is in agreement with that obtained in Eq. (14~) of Ref. [3] in which the 3-gravitino term appears. This term does not appear in the expression computed by Pilati in Ref. [13]. On the other hand, from the comparison between our results and those of Ref. [3], we can see that there are some differences of factors in the final expressions. The reasons for such differences are as follows: first, our Hamiltonian differs by an overall factor from that used in [3]; second, we work with the gamma matrices r,, = 2oab and not with ~~~~ and finally, our definitions of the 2-form Ci, the 3-form Q x , and the momentum # (Eq. (4.22)), also differ by factors from the analogous definitions given in [3].
6. DIRAC
BRACKETS
As it was already commented, in the first order CCF all the primary constraints (2.1) are of second class. It is well known [9, 10, 241 that the second class constraints cannot be interpreted as symmetry generators hence they must be eliminated from the theory. This is done by using the Dirac brackets which are obtained from the second class constraints 0, by means of the definition
CF,Gl* = [F, Gl-
[F, 52,1C="[sZ,,,
G],
where C”‘[Q,,, Q,] = dzB. Here the indices C, LI, 6’ are compound all the possible second class constraints. The important properties of the Dirac brackets are: (i)
(6.1)
and Q, denotes
If one of the functions F or G is first class, then
[F,G]*z[F,Gl.
(6.2)
SECOND
In particular
ORDER
for the Hamiltonian
CANONICAL
225
FORMALISM
J? we have
[F, 2’3*
z [F, 21.
(6.3)
This means that the same equations of motion are obtained by using the Poisson or the Dirac brackets. Thus the rate of change in time of any functional F of the canonical variables is also given by P= [F, 2-J*. (ii)
For any functional
(6.4)
F of the canonical variables it is [O,, F]* =O.
(6.5)
Therefore we can set Q, = 0 either before or after evaluating the Dirac brackets. To compute the Dirac brackets (6.1) we must first obtain the set QL of second class constraints by considering the restriction to C of all the constraints (2.1). The second class constraint (2.la) is easily eliminated in the second order formalism. Furthermore, in this formalism Eq. (2.1 b) which contains the spin connection mab and hence the time derivative of the vierbien, is.no longer a constraint. When we take such an equation strongly equal to zero, it allows us to write the expression for the momenta rc,. Thus the set of second class constraints are given by
Q,(x) = [Q,(x), ...?Q,(x)1 = (x*Qz, x*@?,x*‘v,, x*ya,x*@,,4), x *@(B,?x*‘Pa2X*?n)
(6.6)
where Q,, = 7T,-4&y’&@
g’12
(6.7a)
azk
g”’
(6.7b)
= lTk - 4y,y’5’Eiik
Q3 = 6- 6iXy, g”’
(6.7~)
Q, = 0 + 6iy,lg1’2
(6.7d)
52, = IZ,,, + 6( U, + xTii[;
+ a iB[jyk~jsuk)
Q2, = IT{*, + 6( W, + $Xrjkti~iik
g’/*
- $iAryk(jeiik)
(6.7e) g”’
I
(6.7f)
Q,, = YJX)
(6.39
Q,u = E,(X).
(6.7h)
Using Eqs. (3.3), the Poisson brackets between the constraints computed and they are given by
(6.7) can be
226
FOUSSATS
AND
ZANDRON
CQ~i(xhQ*j(Y)I = -8g1’2y,ykEij/c83(X, .Y) [Q,i(x),
Q,(y)]
= -3ig”2(~y,r’k-B~‘~k)&ijkS3(X,
[Q,,(X),
Q,(y)]
=3g”2(XT’k-iAgiyk)&ijkS3(X,
[Q,(x),
Q,(y)]
=
[Q,(x),
l22,(y) -J = 3g1’2[ir,k&~k63(x,
CQ,(X)?
Q,(Y)1= 1w’2Y.~3b, y)
[Q,(x), G,(y)]
-6g”‘[J-b3(x,
(6.8a) y) J’)
(6.8b) (6.8c)
y)
(6.8d)
y)
(6.8e)
= 3ig1’2[jy&ijkd3(x,
(6.8f) y)
[Q&J, %a(~)1= CQ,tx),Q,,b)l = 6g”*d3k
t6.W
Y)
(6.8h) Poisson
and the corresponding brackets for the adjoint Dirac spinor. All the 0th brackets vanish. From the matrix elements (6.8) and by straightforward calculation the elements of the inverse matrix P” can be computed and we find cf = c; = &g-‘12y,yjyi (6.9a) c34 = c43 = & g - 1/2yI (6.9b) cy= -(-;5=cf+ -Cy=&‘wn, (6.9~) cy=
-qy=c;l,‘=
CA;= -c;;=c;;= c;l=
- pJI =
czg= -cd”= Cfi=
c;3
-C~:=~g-“*n,(y,n+By5~i)
(6.9d)
-Cff=
(6.9e)
= _
C;4
-~g-1’*n,y,(y,n+iA5i) = & g- Wn,yiti
cj13= -Cjf4=
(6.9f)
-&g-l12n,y,yi~i
(6%) (6.9h)
-C~~=$g-1i2n,n,[aX(B~,-iA)T,S,+~iykS,le~k.
Using the definition of the Dirac brackets (6.1) and the result (6.9) we arrive at the field-field Dirac brackets, (6.10a) lIlitxL
u~(Y)l*
=
cA7(x)63(x3
Y)
[5itx)3 wa(Y)l* = cAf(x)s3(x~Y) CG), w)l* = -c43w3(x, Y) C4xL u,(Y)l* = cw~3(x~ Y) [4x), wa(Y)l* = c3xv3(x~ Y) CA(x),gal* = C%) d3(x,Y) [B(x), wab)l* = C:*W3(x, v) CU,b), ~bb)l* = C~W3(X> Y).
(6.10b) (6.10~) (6.10d) (6.1Oe) (6.10f)
(6.W) (6.10h) (6.1Oi)
227
SECONDORDERCANONICALFORMALISM
Analogously
the momentum-field
Dirac brackets are
Cn,‘Cx),Lbj(Y)l* = Cn,‘tx), Lbj(Y)l [n,‘(x), p(y)]* = ~&JJ’rkiJ3(x, y) L-n,‘(-~h ay11* = -gwd [n,‘(x), A(Y = 0 Cn,‘(x), Q)l* [II,‘(x),
[mu”,
(6.11b) (6.11~)
+ n,yiy,)d3(x, y)
(6.11d) (6.1 le)
=0
Ub(y)]*=
wb(y)]*
(6.1 la)
-nb(U,L,‘-n,U’-i;lrak5,&lik
=
+
+iB
-n”(
&Lo’-
-
(6.1 If)
y) ncr w’+
$iA[jiya~k&ijk)63(x,
Finally the Dirac bracket between the momentum
Xr,,l,&‘ik
(6.1&J
y).
components
is
(6.12) The Dirac brackets (6.10a, d), (6.11b, c), and (6.12) can be confronted with those obtained in Refs. [ 13, lo] for pure supergravity and for gravity coupled to the spin i field.
7. CONCLUSIONS
We have shown that both formalisms, the CCF and the CVF can be related. This relation is non-trivial and it is obtained by remarking that the form bracket detinition introduced in the CCF and the Poisson brackets used in the CVF may be connected through an integral relationship. For the particular case of the W-Z@ SUGRA coupled system the method to obtain the second order CVF starting from the first order CCF was explicitly developed. We found the expression for the spin connection and we showed that the contorsion tensor depends on the fermionic fields of the model. Moreover, starting from the total Hamiltonian provided by the CCF containing the set of second-class primary constraints, it was possible to arrive at the Hamiltonian as generator of time evolutions, in the CVF. By following this procedure it is possible to show that 2 is weakly zero because it is given by pieces, each one of them is precisely one of the inhomogeneous equations of motion for the supergravitational field with source, plus terms containing constraints. Subsequently, from this expression and by straightforward calculation the Hamiltonian was written in terms of the first-class constraints which closes the supersymmetric
228
FOUSSATS
AND
ZANDRON
constraint algebra. In particular, the piece Y&, = M,,xO (generators of the local Lorentz group) was obtained in a natural way. On the other hand, the second-class primary constraints obtained in the CCF were analyzed. We have shown that this analysis performed in the framework of the CCF is simpler than that given by starting directly from the canonical vierbein formalism. We also found the Dirac brackets based on the second class constraints for all the fields which take place in the model. For the pure supergravity sector (without W-Z fields) the results obtained in this work can be confronted with others previously obtained and the agreements can be established. The above results show how the CCF provides a useful guidance in writing the CVF in a more simple and compact way.
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